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Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
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Cited by 53 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
The GirardReynolds isomorphism (second edition
 Theoretical Computer Science
, 2004
"... polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfi ..."
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Cited by 5 (0 self)
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polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by wellbehaved models. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version.